rm(list = ls())
library(tidyverse)

Simple Linear Regression

data: lexical decision task

  • lexical decision task: respondents have to determine whether displayed item is a “word” (vs. gibberish “not-word”)
  • data taken from the English Lexicon Project
  • here: have a look at response time for actual words
ldt <- read_csv("../data/winter-elp.csv", show_col_types = FALSE)
glimpse(ldt)
## Rows: 12
## Columns: 3
## $ word <chr> "thing", "life", "door", "angel", "beer", "disgrace", "kitten", "…
## $ freq <dbl> 55522, 40629, 14895, 3992, 3850, 409, 241, 238, 66, 32, 4, 4
## $ rt   <dbl> 621.77, 519.56, 507.38, 636.56, 587.18, 705.00, 611.26, 794.35, 7…
  • hypothesis: the more frequent the word, the easier to decide (the shorter the time needed to decide)
  • plot response time against frequency to visualise effect:
ldt %>% 
  ggplot(aes(x = freq, y = rt, label = word)) +
  geom_point() +
  geom_text(nudge_y = -10)

some terminology:

  • \(y\) is the dependent or explained variable (response, outcome)
  • \(x\) is the independent or explanatory variable (predictor, regressor)

log-scaling:

  • looks indeed like more frequent words elicit shorter response times
  • in general, it is useful to transform frequencies to a log-scale
  • log-frequency: “order” or “magnitude” of frequency
ldt %>%
  ggplot(aes(x = log(freq), y = rt, label = word)) + 
  geom_point() +
  geom_text(nudge_y = -10) +
  geom_smooth(formula = y ~ x, method = 'lm', se = FALSE)

  • now this looks like a linear dependency, indicated by the line of best fit

line of best fit

  • how do we get the line of best fit?
  • minimise sum of squared errors (= vertical difference between points and line)
  • magic very super-complicated formula (/s) yields slope and intercept of linear regression:

\[ \text{rt} \approx b_0 + b_1\cdot \log(\text{freq}) \]

(model <- lm(rt ~ log(freq), data = ldt))
## 
## Call:
## lm(formula = rt ~ log(freq), data = ldt)
## 
## Coefficients:
## (Intercept)    log(freq)  
##      870.91       -30.52
  • note that above notation includes the intercept, it is equivalent to
lm(rt ~ 1 + log(freq), data = ldt)
## 
## Call:
## lm(formula = rt ~ 1 + log(freq), data = ldt)
## 
## Coefficients:
## (Intercept)    log(freq)  
##      870.91       -30.52

interpretation:

  • intercept: a word with \(\log(\text{freq}) = 0\) (\(\text{freq}=1\)) elicits a response time of \(871ms\)
  • slope: every unit step on the \(x\)-axis leads to \(30ms\) less response time

Q: How long would you expect the response time to be for a word that appears 100,000 times? How long for one that appears \(10^{15}\) times? How long for one that appears 0 times? How long for one that appears -1000 times?

residuals and assumptions

  • the residuals are the difference between the fitted values and the actually observed ones
  • the complete model with residuals \(\varepsilon\) looks like this:

\[ \text{rt} = b_0 + b_1\cdot \log(\text{freq}) + \varepsilon \]

  • we can directly access the residuals via
model$residuals
##         1         2         3         4         5         6         7         8 
##  84.28883 -27.45269 -70.25887  18.73354 -31.75190  17.63728 -92.24566  90.46203 
##         9        10        11        12 
## -17.99428  44.74122 -65.09475  48.93525
  • fitting a linear model presupposes two important assumptions regarding these residuals:
    1. normally distributed and
    2. homoscedastic (= equal variance across the \(x\)-axis)
  • checking these assumptions is called “running the model diagnostics”
  • it is usually done visually:
plot(model)

  • the first two plots are the most important ones

    1. the first one is suited for checking homoscedasticity
    2. the second one for checking normality

  • here it looks like a good model, since both assumptions seem to be met

  • NB: we can also check most important descriptive figures of distribution of residuals in model summary:

summary(model)
## 
## Call:
## lm(formula = rt ~ log(freq), data = ldt)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -92.246 -40.088  -0.179  45.790  90.462 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   870.90      40.42  21.548 1.03e-09 ***
## log(freq)     -30.52       5.76  -5.299 0.000348 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 63.34 on 10 degrees of freedom
## Multiple R-squared:  0.7374, Adjusted R-squared:  0.7111 
## F-statistic: 28.08 on 1 and 10 DF,  p-value: 0.0003482

goodness of fit: \(R^2\)

  • how well does the model describe the data?
  • one possibility is to check the sum of squared errors (SSE), where error means residual
  • SSE is minimized for the found parameters, but is the linearity assumption reasonable at all?
  • idea: compare SSE to a null model that does not take into account the predictor:
(model.null <- lm(rt ~ 1, data = ldt))
## 
## Call:
## lm(formula = rt ~ 1, data = ldt)
## 
## Coefficients:
## (Intercept)  
##       679.9
  • this is obviously just the mean of the dataset
mean(ldt$rt)
## [1] 679.9167
  • we can calculate SSE via
sum(model$residuals ** 2)
## [1] 40114.97
sum(model.null$residuals ** 2)
## [1] 152743.2
  • we see: SSE of null model is way higher than SSE of full model
  • a standard measure for the goodness of fit is the ratio of explained variance in comparison to the variance of the null model
  • a bit of arithmetic shows that this is equivalent to

\[R^2:=1-\dfrac{\text{SSE}_\text{model}}{\text{SSE}_\text{null}}\]

1 - sum(model$residuals ** 2) / sum(model.null$residuals ** 2)
## [1] 0.7373698
  • for studies in linguistics, such a high \(R^2\) looks pretty good
  • (NB we’re not doing rocket science here)
  • a bit more arithmetic shows that \(R^2\) is the squared correlation coefficient:
cor(log(ldt$freq), ldt$rt) ** 2
## [1] 0.7373698
  • we can get this number straight away using
summary(model)$r.squared
## [1] 0.7373698

goodness of fit: RSE (Residual Standard Error)

  • measure for how much a prediction is typically off
summary(model)$sigma
## [1] 63.33638
# just the mean of the squared residuals (taking into account the actual degrees of freedom)
sqrt(sum(model$residuals ** 2) / (length(model$residuals) - length(model$coefficients)))
## [1] 63.33638
  • i.e. on average, we assume we’re off by about \(63ms\)

coefficients & predictions

model$coefficients
## (Intercept)   log(freq) 
##   870.90539   -30.52068
  • predictions on original data:
model$fitted.values
##        1        2        3        4        5        6        7        8 
## 537.4812 547.0127 577.6389 617.8265 618.9319 687.3627 703.5057 703.8880 
##        9       10       11       12 
## 743.0343 765.1288 828.5947 828.5947
  • equivalent:
predict(model, ldt)
##        1        2        3        4        5        6        7        8 
## 537.4812 547.0127 577.6389 617.8265 618.9319 687.3627 703.5057 703.8880 
##        9       10       11       12 
## 743.0343 765.1288 828.5947 828.5947
  • predictions on some other data:
explanatory_data <- tibble(
  freq = c(-500, 1.2, 10000, 10**100)
)
predict(model, explanatory_data)
## Warning in log(freq): NaNs produced
##          1          2          3          4 
##        NaN   865.3408   589.7995 -6156.7408

significance

  • \(R^2\) is an effect size: how much variance is explained
  • how to assess whether model fit is significant?
summary(model)
## 
## Call:
## lm(formula = rt ~ log(freq), data = ldt)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -92.246 -40.088  -0.179  45.790  90.462 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   870.90      40.42  21.548 1.03e-09 ***
## log(freq)     -30.52       5.76  -5.299 0.000348 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 63.34 on 10 degrees of freedom
## Multiple R-squared:  0.7374, Adjusted R-squared:  0.7111 
## F-statistic: 28.08 on 1 and 10 DF,  p-value: 0.0003482
  • look at p-value of each coefficient (and significance coding next to it)
  • look at p-value at very bottom: is the model significant as a whole?

standardisation and centring

  • log-scaling often useful, changes model fit

  • other transformation possible, e.g. polynomial

  • linear transformations, on the other hand, do not change model fit

  • it is often recommended to centre or standardise variables

    • centring = subtract mean
    • standardising = subtract mean, divide by std. dev.

  • goal: resulting variables are more easily interpretable

ldt <- ldt %>%
  mutate(log_freq = log(freq),
         log_freq_z = (log_freq - mean(log_freq)) / sd(log_freq),
         rt_z = (rt - mean(rt)) / sd(rt))
  • NB: “z” stands for z-score (cf. normally distributed variables)
ldt %>%
  ggplot(aes(x = log_freq, y = rt, label = word)) + 
  geom_point() +
  geom_text(nudge_y = -10) +
  geom_smooth(formula = y ~ x, method = 'lm', se = FALSE)

ldt %>%
  ggplot(aes(x = log_freq_z, y = rt, label = word)) + 
  geom_point() +
  geom_text(nudge_y = -10) +
  geom_smooth(formula = y ~ x, method = 'lm', se = FALSE)

ldt %>%
  ggplot(aes(x = log_freq_z, y = rt_z, label = word)) + 
  geom_point() +
  geom_text(nudge_y = -.1) +
  geom_smooth(formula = y ~ x, method = 'lm', se = FALSE)

  • they all look exactly the same, just on different scales
  • and indeed the models are equally good:
model1 <- lm(rt ~ log_freq, data = ldt)  # same as original model
model2 <- lm(rt ~ log_freq_z, data = ldt)
model3 <- lm(rt_z ~ log_freq_z, data = ldt)
summary(model1)$r.squared
## [1] 0.7373698
summary(model2)$r.squared
## [1] 0.7373698
summary(model3)$r.squared
## [1] 0.7373698
  • however, the coefficients are different:
model1$coefficients
## (Intercept)    log_freq 
##   870.90539   -30.52068
model2$coefficients
## (Intercept)  log_freq_z 
##    679.9167   -101.1876
model3$coefficients
##   (Intercept)    log_freq_z 
##  1.121728e-16 -8.587024e-01
  • interpretation for intercept:
    1. a word with \(\log(\text{freq}) = 0\) (\(\text{freq}=1\)) elicits a response time of \(871ms\)
    2. an “averagely frequent” word elicits a response time of \(680ms\)
    3. 0
  • interpretation for slope:
    1. every unit step on the \(x\)-axis leads to \(30ms\) less response time
    2. stepping one std. dev. of \(\log\text{freq}\) to the right decreases rt by \(101ms\)
    3. correlation coefficient

tidy models

  • output format of lm() and summary() not straightforward
  • use tibbles and package broom for tidy formats
  • look at model as a whole
broom::glance(model)
## # A tibble: 1 × 12
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <dbl>  <dbl> <dbl> <dbl>
## 1     0.737         0.711  63.3      28.1 0.000348     1  -65.7  137.  139.
## # … with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
  • look at coefficients
broom::tidy(model)
## # A tibble: 2 × 5
##   term        estimate std.error statistic       p.value
##   <chr>          <dbl>     <dbl>     <dbl>         <dbl>
## 1 (Intercept)    871.      40.4      21.5  0.00000000103
## 2 log(freq)      -30.5      5.76     -5.30 0.000348
  • look at each fitted observation
broom::augment(model)
## # A tibble: 12 × 7
##       rt `log(freq)` .fitted   .hat .sigma .cooksd .std.resid
##    <dbl>       <dbl>   <dbl>  <dbl>  <dbl>   <dbl>      <dbl>
##  1  622.       10.9     537. 0.263    58.2 0.430        1.55 
##  2  520.       10.6     547. 0.240    65.9 0.0391      -0.497
##  3  507.        9.61    578. 0.176    61.6 0.160       -1.22 
##  4  637.        8.29    618. 0.118    66.4 0.00660      0.315
##  5  587.        8.26    619. 0.116    65.8 0.0187      -0.533
##  6  705         6.01    687. 0.0838   66.5 0.00387      0.291
##  7  611.        5.48    704. 0.0883   58.5 0.113       -1.53 
##  8  794.        5.47    704. 0.0884   58.8 0.109        1.50 
##  9  725.        4.19    743. 0.119    66.5 0.00617     -0.303
## 10  810.        3.47    765. 0.148    64.8 0.0508       0.765
## 11  764.        1.39    829. 0.280    61.7 0.285       -1.21 
## 12  878.        1.39    829. 0.280    63.9 0.161        0.910

Categorical Predictors

  • until now, we were only looking at metric variables
  • now: categorical predictor (still metric response)
  • what about the kind of words
  • here: stopwords (easily accessible)
# whole data set
ldt.c <- read_csv("../data/winter-elp-complete.csv", show_col_types = FALSE)
stops <- stopwords::stopwords()  # English stopwords
ldt.c <- ldt.c %>% mutate(stop = factor(word %in% stops))
  • stopwords are predominantly very frequent:
ldt.c %>% ggplot(aes(x = stop, y = rt, fill = stop)) +
  geom_boxplot()

ldt.c %>% ggplot(aes(x = log(freq), fill = stop)) +
  geom_density(alpha = .5)

  • and the response variable?
ldt.c %>% ggplot(aes(x = log(freq), y = rt, color = stop)) +
  geom_point()

  • remember easiest form of linear model: \[ y = b_0 + b_1\cdot x + \varepsilon \]

  • obviously, does not work if \(x\) is categorical

  • however, we can just use a dummy variable, coding the factor as a metric variable

  • here: treatment coding: we replace one of the levels as 0 (the reference level), the other one as 1

model.stop <- lm(rt ~ stop, data = ldt.c)
  • i.e. the prediction for reference level (here: FALSE) is \(770ms\)
  • the prediction for stopwords is \(605ms\)
  • and these are obviously just the mean values:
ldt.c %>% group_by(stop) %>% summarise(mean_rt = mean(rt))
## # A tibble: 2 × 2
##   stop  mean_rt
##   <fct>   <dbl>
## 1 FALSE    770.
## 2 TRUE     605.
summary(model.stop)
## 
## Call:
## lm(formula = rt ~ stop, data = ldt.c)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -355.08  -90.52  -17.29   71.01  897.92 
## 
## Coefficients:
##              Estimate Std. Error  t value Pr(>|t|)    
## (Intercept)  770.0768     0.6838 1126.196   <2e-16 ***
## stopTRUE    -165.1377    19.4213   -8.503   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 124.3 on 33073 degrees of freedom
## Multiple R-squared:  0.002181,   Adjusted R-squared:  0.002151 
## F-statistic:  72.3 on 1 and 33073 DF,  p-value: < 2.2e-16
plot(model.stop)

Multiple Regression

  • up until now: just one predictor
  • multiple regression: use several predictors
  • here: word length (measured in number of phonemes) might also play a role
  • clearly, shorter words are read more quickly, yielding a faster response
  • NB: frequent words tend to be shorter! conflation of effects

two simple linear regressions

\[ \text{rt} = b_0 + b_1\cdot \log(\text{freq}) + \varepsilon \]

ldt.c %>% ggplot(aes(x = log(freq), y = rt)) + 
  geom_point() +
  geom_smooth(formula = y ~ x, method = 'lm', se = FALSE)

model.freq <- lm(rt ~ log(freq), data = ldt.c)
broom::tidy(model.freq)
## # A tibble: 2 × 5
##   term        estimate std.error statistic p.value
##   <chr>          <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)    907.      1.09       828.       0
## 2 log(freq)      -37.5     0.262     -143.       0
broom::glance(model.freq)
## # A tibble: 1 × 12
##   r.squared adj.r.squared sigma statistic p.value    df   logLik     AIC     BIC
##       <dbl>         <dbl> <dbl>     <dbl>   <dbl> <dbl>    <dbl>   <dbl>   <dbl>
## 1     0.383         0.383  97.7    20566.       0     1 -198475. 396956. 396981.
## # … with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
plot(model.freq)

\[ \text{rt} = b_0 + b_1\cdot \text{length} + \varepsilon \]

ldt.c %>% ggplot(aes(x = length, y = rt)) + 
  geom_point() +
  geom_smooth(formula = y ~ x, method = 'lm', se = FALSE)

model.length <- lm(rt ~ length, data = ldt.c)
broom::tidy(model.length)
## # A tibble: 2 × 5
##   term        estimate std.error statistic p.value
##   <chr>          <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)    572.      1.82       314.       0
## 2 length          29.8     0.260      115.       0
broom::glance(model.length)
## # A tibble: 1 × 12
##   r.squared adj.r.squared sigma statistic p.value    df   logLik     AIC     BIC
##       <dbl>         <dbl> <dbl>     <dbl>   <dbl> <dbl>    <dbl>   <dbl>   <dbl>
## 1     0.284         0.284  105.    13112.       0     1 -200949. 401905. 401930.
## # … with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
plot(model.length)

multiple predictors

continuous + continuous

  • now: take both as explanatory variables \[ \text{rt} = b_0 + b_1\cdot \log(\text{freq}) + b_2\cdot \text{length} + \varepsilon \]
summary(lm(rt ~ log(freq) + length, data = ldt.c))
## 
## Call:
## lm(formula = rt ~ log(freq) + length, data = ldt.c)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -410.24  -60.52   -9.71   48.01  705.53 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 748.4261     2.1761  343.92   <2e-16 ***
## log(freq)   -29.5418     0.2579 -114.54   <2e-16 ***
## length       19.4583     0.2377   81.86   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 89.09 on 33072 degrees of freedom
## Multiple R-squared:  0.4873, Adjusted R-squared:  0.4873 
## F-statistic: 1.572e+04 on 2 and 33072 DF,  p-value: < 2.2e-16
  • this is obviously not the same as fitting two separate models
  • reasonable interpretation for standardised (or at least centred) variables
ldt.c <- ldt.c %>% mutate(log_freq = log(freq),
                          log_freq_z = (log_freq - mean(log_freq) / sd(log_freq)),
                          length_c = (length - mean(length)))
model.freq.length <- lm(rt ~ log_freq_z + length_c, data = ldt.c)
summary(model.freq.length)
## 
## Call:
## lm(formula = rt ~ log_freq_z + length_c, data = ldt.c)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -410.24  -60.52   -9.71   48.01  705.53 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 825.0849     0.6872 1200.58   <2e-16 ***
## log_freq_z  -29.5418     0.2579 -114.54   <2e-16 ***
## length_c     19.4583     0.2377   81.86   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 89.09 on 33072 degrees of freedom
## Multiple R-squared:  0.4873, Adjusted R-squared:  0.4873 
## F-statistic: 1.572e+04 on 2 and 33072 DF,  p-value: < 2.2e-16
  • intercept: an averagely frequent word with average length elicits a response time of \(825ms\)
  • slope 1: increase of 1 std. dev. of magnitude of frequency (c.p.): decrease of \(30ms\)
  • slope 2: increase of 1 phoneme (c.p.): increase of \(20ms\)
broom::glance(model.freq.length)
## # A tibble: 1 × 12
##   r.squared adj.r.squared sigma statistic p.value    df   logLik     AIC     BIC
##       <dbl>         <dbl> <dbl>     <dbl>   <dbl> <dbl>    <dbl>   <dbl>   <dbl>
## 1     0.487         0.487  89.1    15717.       0     2 -195424. 390856. 390889.
## # … with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
broom::tidy(model.freq.length)
## # A tibble: 3 × 5
##   term        estimate std.error statistic p.value
##   <chr>          <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)    825.      0.687    1201.        0
## 2 log_freq_z     -29.5     0.258    -115.        0
## 3 length_c        19.5     0.238      81.9       0

categorical + continuous

  • similarly, categorical predictors can be included
model.length.stop <- lm(rt ~ length_c + stop, data = ldt.c)
summary(model.length.stop)
## 
## Call:
## lm(formula = rt ~ length_c + stop, data = ldt.c)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -306.31  -75.69  -13.93   61.26  768.37 
## 
## Coefficients:
##             Estimate Std. Error  t value Pr(>|t|)    
## (Intercept) 769.9417     0.5792 1329.369  < 2e-16 ***
## length_c     29.7198     0.2604  114.136  < 2e-16 ***
## stopTRUE    -56.2101    16.4778   -3.411 0.000647 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 105.3 on 33072 degrees of freedom
## Multiple R-squared:  0.2842, Adjusted R-squared:  0.2841 
## F-statistic:  6564 on 2 and 33072 DF,  p-value: < 2.2e-16
  • an averagely long word, which is not a stopword, elicits a reaction time of \(770ms\)
  • visualisation via additional library
ldt.c %>% ggplot(aes(length_c, rt, color = stop)) +
  geom_point(alpha = .5) +
  moderndive::geom_parallel_slopes(se = F)

categorical + categorical

  • one mean for each category

interactions

continuous * continuous

  • twisted planes

categorical * continuous

  • different slopes for different categories
model.length.stop.inter <- lm(rt ~ length_c * stop, data = ldt.c)
summary(model.length.stop.inter)
## 
## Call:
## lm(formula = rt ~ length_c * stop, data = ldt.c)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -306.31  -75.69  -13.93   61.27  768.35 
## 
## Coefficients:
##                    Estimate Std. Error  t value Pr(>|t|)    
## (Intercept)        769.9417     0.5792 1329.390   <2e-16 ***
## length_c            29.7246     0.2604  114.147   <2e-16 ***
## stopTRUE          -163.3215    75.9931   -2.149   0.0316 *  
## length_c:stopTRUE  -29.2654    20.2691   -1.444   0.1488    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 105.3 on 33071 degrees of freedom
## Multiple R-squared:  0.2842, Adjusted R-squared:  0.2841 
## F-statistic:  4377 on 3 and 33071 DF,  p-value: < 2.2e-16
ldt.c %>% ggplot(aes(length_c, rt, color = stop)) +
  geom_point(alpha = .5) +
  geom_smooth(method = 'lm', se = F)
## `geom_smooth()` using formula 'y ~ x'

categorical * categorical

  • full four-way split

collinearity

  • collinearity of predictors means that at least one predictor can be described as a linear combination of the other predictors
  • let’s measure word length by number of characters:
ldt.c <- ldt.c %>% mutate(length_char = str_length(word),
                          length_char_c = length_char - mean(length_char))
ldt.c %>% ggplot(aes(x = length, y = length_char_c)) + geom_point() +
  geom_smooth(formula = y ~ x, method = 'lm', se = FALSE)

  • clearly the two variables measuring length are somewhat collinear to one another!
  • what does the model do?
model.length.length <- lm(rt ~ length_c + length_char_c, data = ldt.c)

rbind(
  broom::glance(model.length),
  broom::glance(model.length.length)
)
## # A tibble: 2 × 12
##   r.squared adj.r.squared sigma statistic p.value    df   logLik     AIC     BIC
##       <dbl>         <dbl> <dbl>     <dbl>   <dbl> <dbl>    <dbl>   <dbl>   <dbl>
## 1     0.284         0.284  105.    13112.       0     1 -200949. 401905. 401930.
## 2     0.302         0.302  104.     7159.       0     2 -200523. 401054. 401087.
## # … with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
rbind(
  broom::tidy(model.length),
  broom::tidy(model.length.length)
)
## # A tibble: 5 × 5
##   term          estimate std.error statistic   p.value
##   <chr>            <dbl>     <dbl>     <dbl>     <dbl>
## 1 (Intercept)      572.      1.82      314.  0        
## 2 length            29.8     0.260     115.  0        
## 3 (Intercept)      770.      0.572    1347.  0        
## 4 length_c          12.9     0.629      20.5 1.28e- 92
## 5 length_char_c     17.2     0.586      29.4 1.53e-187

  • adding another predictor has not given us much gain in \(R^2\) (also look at adjusted \(R^2\))

  • it did however change the coefficient for the length drastically, changing the interpretation!

  • we can use variance inflation factors (vif) to check if we are dealing with collinearity

  • vif should be close to one; it should start worrying you when it’s above 5

car::vif(model.length.stop)
## length_c     stop 
## 1.003366 1.003366
car::vif(model.length.length)
##      length_c length_char_c 
##      6.008984      6.008984

model diagnostics

  • interpretation of diagnostic plots as above
plot(model.length.stop)